# Ditrigonal icosahedron

Ditrigonal icosahedron
Rank3
TypeSemiuniform
Notation
Bowers style acronymDitti
Elements
Faces20 propeller tripods
Edges30+30
Vertices20
Vertex figureConcave triambus
Measures (edge lengths 1, ${\displaystyle {\frac {3+{\sqrt {5}}}{2}}}$)
Edge length ratio${\displaystyle {\frac {3+{\sqrt {5}}}{2}}\approx 2.61803}$
Circumradius${\displaystyle {\frac {{\sqrt {3}}+{\sqrt {15}}}{4}}\approx 1.40126}$
Central density0
Number of external pieces60
Level of complexity3
Related polytopes
ArmyDoe
RegimentDitti
DualDitrigonal icosahedron
ConjugateDitrigonal icosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Euler characteristic–20
Schläfli type{6,6}
OrientableYes
Genus11
Properties
SymmetryH3, order 120
Flag orbits2
ConvexNo
NatureTame
History
Discovered byEdmond Hess
First discovered1877

The ditrigonal icosahedron, or ditti, also called the excavated dodecahedron, is a noble self-dual semi-uniform polyhedron. It consists of 20 propeller tripods, with six joining at a vertex. It has two lengths of edges, but it has no degrees of variance; the longer edges are ${\displaystyle {\frac {3+{\sqrt {5}}}{2}}\approx 2.61803}$ times the length of the shorter edges. The longer edges are also the edges of a great stellated dodecahedron, while the shorter edges are the edges of a dodecahedron. It is a stellation of the icosahedron.

The ditrigonal icosahedron is a two-orbit polyhedron, and also the only two-orbit polyhedron which is also noble. If one counts conjugacies as symmetries, the ditrigonal icosahedron is a regular polyhedron. Abstractly this polyhedron is a quotient of the order-6 hexagonal tiling.

Extending the shorter edges while keeping the face planes results in a great icosahedron, while contracting the longer edges while keeping the face planes results in an icosahedron. Relatedly, it can be constructed by antitruncating the icosahedron or hypertruncating the great icosahedron until the resulting pentagons or pentagrams meet, causing the polyhedron to be exotic. Removing the pentagons or pentagrams (which form the faces of a dodecahedron or great stellated dodecahedron) results in the ditrigonal icosahedron.

This polyhedron appears as the vertex figure of the uniform invertidodecahedronary hecatonicosachoron.